Anonymous Functions/Operators/Hyperators

Anonymous functions and operators (AFOs) are a one-line grouping of one or more statements all enclosed in braces such as {(+⌿⍵)÷≢⍵}. This syntax is useful for one-line functions and operators to complement the existing definition types of user-defined: ∇ Z←avg R, trains: (+⌿ ÷ ≢), and derived: ,∘⍋∘⍋∘,.

These objects may be named as in avg←{(+⌿⍵)÷≢⍵}, and they may be used in place of any APL function (primitive, operator operand, derived, train, user-defined) including within another AFO.

Normally, the statements within the braces execute one by one from left to right, just as in execute. Execution of the AFO terminates on the first occurrence of one of the following:

• A Guard with a TRUE condition,
• A non-Shy value, or
• The rightmost statement.

Function Arguments

To define an anonymous function, use ⍺ as the (optional) left argument, ∇ as the name of the anonymous function (for recursion), and ⍵ as the name of the right argument. For example,

      3{√+/⍺ ⍵*2}4                ⍝ Pythagorean theorem
5
      {s←(+/⍵)÷2 ⋄ √×/s-0,⍵}3 4 5 ⍝ Heron's formula for triangle area
6

Operator Operands

To define an anonymous operator, use the above special names along with ⍺⍺ as the name of the left operand, ∇∇ as the name of the operator (for recursion), and ⍵⍵ as the name of the right operand. If neither ⍺⍺ nor ⍵⍵ appears as a token (outside of character constants) between the braces, then the object is a function, not an operator. For example,

f←{∘.⍺⍺⍨⍳⍵}
=f 4 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1	⌈f 4 1 2 3 4 2 2 3 4 3 3 3 4 4 4 4 4	*f 4 1  1  1   1 2  4  8  16 3  9 27  81 4 16 64 256	≤f 4 1 1 1 1 0 1 1 1 0 0 1 1 0 0 0 1

Ambivalent AFOs

User-defined functions/operators allow you to specify in their headers that the left argument is optional by enclosing it in braces, as in ∇ Z←{L} foo R. This behavior is also available to AFOs by including a statement that assigns a value to ⍺. If ⍺ does not have a value when that statement is encountered, the statement is executed; otherwise, that entire statement is ignored including any side effects. This behavior obviates using 0=⎕NC '⍺' to test for a value in ⍺.

For example,

      f←{⍺←2 ⋄ ⍺√⍵}
      f 16
4
      3 f 27
3
      f←{⎕←⍺←⍵ ⋄ (⍳⍺)∘.⍺⍺⍳⍵}
      ⌊f 4
4
1 1 1 1
1 2 2 2
1 2 3 3
1 2 3 4
      3⌊f 4
1 1 1 1
1 2 2 2
1 2 3 3

As a consequence of this rule, regardless of whether the AFO is called monadically or dyadically, any second or subsequent statements that assign a value to ⍺ are always ignored.

Valences

An AFO may be

• Ambivalent (must be called with either one or two arguments),
• Dyadic (must be called with two arguments),
• Monadic (must be called with one argument), or
• Niladic (must be called with no arguments),

depending upon which of the special symbols are present in its definition.

Disregarding special symbols inside of character constants

• For anonymous functions (and operators):
  • If ⍺← appears as a sequence of tokens, then the object is an ambivalent (derived) function,
  • Otherwise, if ⍺ appears as a token, the object is a dyadic (derived) function,
  • Otherwise, if ⍵ appears as a token, the object is a monadic (derived) function,
  • Otherwise, if neither ⍺ nor ⍵ appears as a token, the object is a niladic (derived) function.
• For anonymous operators:
  • If ⍵⍵ appears as a token, the operator is dyadic (must be called with a left and right operand),
  • Otherwise, it is monadic (must be called with a left operand only).

For the moment, the case of a niladic derived function from either a monadic or dyadic operator signals a SYNTAX ERROR.

For example,

      {⍵}2
2
      1{⍵}2
VALENCE ERROR
      1{⍵}2
       ^
      1{⍺+⍵}2
3
      {⍺+⍵}2
VALENCE ERROR
      {⍺+⍵}2
      ∧
      {⍺←2 ⋄ ⍺+⍵}2  ⍝ Ambivalent function, monadic with default left argument 2
4
      3{⍺←2 ⋄ ⍺+⍵}2 ⍝ Ambivalent function, dyadic with left argument 3
5
      {⍳3}          ⍝ A three-element simple vector
1 2 3
      {⍳3}23        ⍝ A two-element nested vector
 1 2 3  23
      12{⍳3}23      ⍝ A three-element nested vector
 12  1 2 3  23
      3{∘.=⍨⍳⍺⍺}
SYNTAX ERROR
      3{∘.=⍨⍳⍺⍺}
       ∧

Guards

Guards are used to test for a condition and execute a statement or not depending upon the value of the conditional expression. This behavior is identical to the control structure

:if Cond ⋄ CondStmt ⋄ :return ⋄ :end

Guards appear as a Boolean-valued APL expression (Cond) followed by a colon, followed by a single APL statement (CondStmt) as in

{... ⋄ Cond:CondStmt ⋄ ....}

which executes CondStmt and terminates the AFO if and only if Cond evaluates to TRUE (1).

For example,

{... ⋄ 0∊⍴⍵:'empty right argument' ⋄ ....}

As many guard statements may appear in an AFO as desired. They are executed in turn until one of them evaluates to TRUE, at which time the statement following the colon is executed and the function terminates with the result of that conditional statement as the result of the AFO.

If Cond does not evaluate to a Boolean-valued scalar or one-element vector, a RANK, LENGTH, or DOMAIN ERROR is signalled, as appropriate.

If the rightmost statement is executed and it is a guard statement whose Cond evaluates to FALSE (0), the AFO terminates with no value; if the result of that AFO is assigned, a VALUE ERROR is signalled.

Shy Results

A "shy" result is any result that doesn't automatically display its value, a simple example of which is L←⍳3: the result is ⍳3 (as evidenced by the extension to 3+L←⍳3), but because it has been assigned to a name, the result doesn't automatically display. Other ways to produce a shy result are to use the Sink syntax ←⍳3, or to call a user-defined function that declares in its header that the result is shy as in ∇ {Z}←foo R.

A shy result propagates up the chain of results such as it does through the Execute primitive. That is, if the last statement executed has a shy result, the result of execute is shy, too.

Typically, ⎕← is used to expose a shy result.

AFOs also may have shy results by virtue of the final result being

• Sinked,
• Assigned to a name, or
• From a shy user-defined function.

For example,

      {←⍳3}
      ⎕←{←⍳3}
1 2 3
      {L←⍳3}
      ⎕←{L←⍳3}
1 2 3

Ordinarily, a shy result doesn't terminate the AFO, as in

      {←⍳3 ⋄ 'abc'}
abc

To force termination, precede the assignment with a guard as in

{⍵=0:←'Zero' ⋄ ⍵>0:←'Positive' ⋄ ←'Negative'} R

which terminates with a shy result in all three cases.

Localization

Unlike user-defined functions, in AFOs all names to which an assignment is made are automatically localized. As a result, a direct assignment inside an AFO cannot affect the value of a name defined higher up in the execution chain unless it is made via a user-defined function or execute as in

      L←⍳9 ⋄ {L←"abc" ⋄ ⍵}23
23
      L
1 2 3 4 5 6 7 8 9
      L←⍳9 ⋄ {⍎'L←"abc"' ⋄ ⍵}23
23
      L
abc

Scoping

TBD

Recursion

TBD

Restrictions

• For the moment, AFOs may be written on one line only.
• As a consequence of the above one-line restriction, AFOs may not contain comments.
• Assignments to any of the special names (∇, ⍵, ⍺⍺, ∇∇, ⍵⍵) except for ⍺ are not allowed; they signal a SYNTAX ERROR.
• Assignments to ⍺ within a guard statement are not allowed; they signal a SYNTAX ERROR.
• None of the special names may be erased, via ⎕EX or otherwise.
• Goto statements (including →, →0, and →⍬) are not allowed; they signal a SYNTAX ERROR.
• Control structures such as :if ... ⋄ ... ⋄ :end are not allowed; they signal a SYNTAX ERROR.